the primary figure shows TMD applications inside the buildings. Picture you need to design a tuned mass damper for a one-story building. Assume no rotation of the horizontal section at the level of the floor. The floor is rigid, and the total mass is concentrated at the level of the floor. Consider the building which m1 = 1,800 kg, k1 = 8,000 N/m, and c1 = 500 Ns/m. The parameters of TMD are md = 1,000 kg, kd = 3,000 N/m, and cd = 100 Ns/m. Use MATLAB to plot the graphs a) Derive the equation of motion using Lagrange method. Consider f(t), c1, and cd equal to zero for part (b)-(e). b) Find natural frequencies and modes of vibration. c) Plot the first and second natural frequencies for different values of kd and explain your observations. d) Find one initial condition to excite the first mode and simulate the response. e) Find one initial condition to excite the second mode and simulate the response. f) Plot the system response for the initial condition x1 = 0.05 m, x1 = 0 m/s, xd = 0.01 m and xd = 0 m/s, and once with c1 = cd = 0 Ns/m, f(t) = 0 and once with c1 = 500 Ns/m, cd = 100 Ns/m, f(t) = 0. g) Assume the seismic force f(t) = 2000sin(wt) applies to the building at the position same as the figure. Plot the amplitude of vibration for x1 and xd versus different frequencies of the excitation and explain your observations. Choose your own w and initial condition x1 = x1 = xd = xd = 0 for part (h) and (i). h) The maximum allowable displacement for this building (x1) is 30 cm. Choose your own kd to fit this requirement, and explain your observations. i) According to the whole building design guide, people lose their balance when the acceleration (1) is more than 20 cm/s2. Hence, choose your own cd to fit this criterion, then explain your observations. Note that you should keep the requirement in (h). ka Ca M1 C1 xa Ma Figure 2. The diagram of building with TMD k 2